Differentiation#
Elementary Forms#
Trigonometric Functions#
\[\begin{split} &\dfrac{d}{dx}\big[\sin x\big] = \cos x & & \dfrac{d}{dx}\big[\cos x\big] = -\sin x \\
&\dfrac{d}{dx}\big[\sec x\big] = \sec x \tan x \qquad & & \dfrac{d}{dx}\big[\csc x\big] = -\csc x \cot x\\
&\dfrac{d}{dx}\big[\tan x\big] = \sec^2 x & & \dfrac{d}{dx}\big[\cot x\big] = -\csc^2 x \\ \end{split}\]
Inverse Trigonometric Functions#
\[\begin{split} &\dfrac{d}{dx}\big[\sin^{-1} x\big] = \dfrac{1}{\sqrt{1-x^2}} \qquad & & \dfrac{d}{dx}\big[\cos^{-1} x\big] = -\dfrac{1}{\sqrt{1-x^2}} \\
&\dfrac{d}{dx}\big[\tan^{-1} x\big] = \dfrac{1}{1+x^2} & & \\ \end{split}\]
Exponential and Logarithmic Functions#
\[\begin{split} &\dfrac{d}{dx}\big[e^x\big] = e^x & & \dfrac{d}{dx}\big[\ln x\big] = \dfrac{1}{x} \\
&\dfrac{d}{dx}\big[b^x\big] = b^x \ln b \qquad & & \dfrac{d}{dx}\big[\ln \lvert x\rvert \big] = \dfrac{1}{x} \\
& & & \dfrac{d}{dx}\big[\log_b x\big] = \dfrac{1}{x\ln b} \\ \end{split}\]
Power Rule#
Power Rule
For real number \(n\)
\[\dfrac{d}{dx}\big[ x^n \big] = n x^{n-1}\]
Algebraic Rules#
\[
\dfrac{d}{dx}\big[f(x)+g(x)\big] = f'(x)+g'(x)
\]
\[
\dfrac{d}{dx}\big[f(x)-g(x)\big] = f'(x)-g'(x)
\]
\[
\dfrac{d}{dx}\big[c\cdot f(x)\big] = c\cdot f'(x)
\]
Product Rule#
\[
\dfrac{d}{dx}\big[f(x)\cdot g(x)\big] = \big[f(x)\big]' \cdot g(x) + f(x)\cdot \big[g(x)\big]'
\]
Quotient Rule#
\[
\dfrac{d}{dx}\left[\dfrac{f(x)}{g(x)}\right] = \dfrac{\big[f(x)\big]' \cdot g(x) - f(x)\cdot \big[g(x)\big]'}{(g(x))^2}
\]
Chain Rule#
\[
\dfrac{d}{dx}\bigg[f(g(x))\bigg] = f'(g(x))\cdot g'(x)
\]